The cosmological constant [Formula: see text] used to be a freedom in Einstein’s theory of general relativity (GR), where one had a proclivity to set it to zero purely for convenience. The signs of [Formula: see text] or [Formula: see text] being zero would describe universes with different properties. For instance, the conformal structure of spacetime directly depends on [Formula: see text]: null infinity [Formula: see text] is a spacelike, null, or timelike hypersurface, if [Formula: see text], [Formula: see text], or [Formula: see text], respectively. Recent observations of distant supernovae have taught us that our universe expands at an accelerated rate, and this can be accounted for by choosing [Formula: see text] in Einstein’s theory of GR. A quantity that depends on the conformal structure of spacetime, especially on the nature of [Formula: see text], is the Bondi mass which in turn dictates the mass loss of an isolated gravitating system due to energy carried away by gravitational waves. This problem of extending the Bondi mass to a universe with [Formula: see text] has spawned intense research activity over the past several years. Some aspects include a closer inspection on the conformal properties, working with linearization, attempts using a Hamiltonian formulation based on “linearized” asymptotic symmetries, as well as obtaining the general asymptotic solutions of de Sitter-like spacetimes. We consolidate on the progress thus far from the various approaches that have been undertaken, as well as discuss the current open problems and possible directions in this area.
On the volume and the Gauss map image of spacelike hypersurfaces in de Sitter space
